Commit 250bd86c authored by linushof's avatar linushof
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Review of introduction to SSM

parent 4ec397da
...@@ -43,19 +43,19 @@ Synthetic choice data is simulated and modeled in cumulative prospect theory to ...@@ -43,19 +43,19 @@ Synthetic choice data is simulated and modeled in cumulative prospect theory to
## Random Processes in Sequential Sampling ## Random Processes in Sequential Sampling
In research on the decision theory, a standard paradigm is the choice between at least two (monetary) prospects. In research on the decision theory, a standard paradigm is the choice between at least two prospects.
Let a prospect be a probability space $(\Omega, \mathscr{F}, P)$. Let a prospect be a probability space $(\Omega, \mathscr{F}, P)$.
$\Omega$ is the sample space $\Omega$ is the sample space
$$\begin{equation} $$\begin{equation}
\omega_i = \{\omega_1, ..., \omega_n\} \in \Omega \Omega = \{\omega_1, ..., \omega_n\}
\end{equation}$$ \end{equation}$$
containing a finite set of possible outcomes, monetary gains and/or losses respectively. containing a finite set of possible outcomes $\omega$, monetary gains and/or losses respectively.
$\mathscr{F}$ is the set of all possible subsets of $\Omega$: $\mathscr{F}$ is the set of all possible subsets of $\Omega$:
$$\begin{equation} $$\begin{equation}
A_i = \{A_1, ..., A_n\} \in \mathscr{F} = \mathscr{P}(\Omega) \mathscr{F} = \{A_1, A_2, ...\} = \mathscr{P}(\Omega)
\; . \; .
\end{equation}$$ \end{equation}$$
...@@ -65,17 +65,18 @@ $$\begin{equation} ...@@ -65,17 +65,18 @@ $$\begin{equation}
P: \mathscr{F} \mapsto [0,1] P: \mathscr{F} \mapsto [0,1]
\end{equation}$$ \end{equation}$$
that assigns each outcome a probability of $0 \leq p(\omega_i) \leq 1$ with $P(\Omega) = 1$ [cf. @kolmogorovFoundationsTheoryProbability1950, pp. 2-3]. that assigns each outcome $\omega$ a probability $0 < p(\omega) \leq 1$ with $P(\Omega) = 1$ [ @kolmogorovFoundationsTheoryProbability1950, pp. 2-3].
In such a choice paradigm, agents are asked to evaluate the prospects and build a preference for either one of them. In such a choice paradigm, agents are asked to evaluate the prospects and build a preference for either one of them.
It is common to make a rather crude distinction between two variants of this evaluation process [cf. @hertwigDescriptionexperienceGapRisky2009]. It is common to make a rather crude distinction between two variants of this evaluation process [cf. @hertwigDescriptionexperienceGapRisky2009].
For decisions from description (DfD), agents are provided a full symbolic description of the prospects. For decisions from description (DfD), agents are provided a full symbolic description of the prospects.
For decisions from experience [DfE; e.g., @hertwigDecisionsExperienceEffect2004], the prospects not described but must be explored by the means of sampling. For decisions from experience [DfE; e.g., @hertwigDecisionsExperienceEffect2004], prospects are not described but must be explored by the means of sampling.
To provide a formal definition of sampling in risky choice, we make use of the mathematical concept of a random variable and start by referring to a prospect as *"risky"* in the case where all $p(\omega_{i}) \neq 1$.
To provide a formal definition of sampling in risky choice, we make use of the mathematical concept of a random variable and start by referring to a prospect as *"risky"* in the case where $p(\omega) \neq 1$ for all $\omega \in \Omega$.
Here, risky describes the fact that if agents would choose a prospect and any of its outcomes in $\Omega$ must occur, none of these outcomes will occur with certainty. Here, risky describes the fact that if agents would choose a prospect and any of its outcomes in $\Omega$ must occur, none of these outcomes will occur with certainty.
It is acceptable to speak of the occurrence of $\omega_{i}$ as a realization of a random variable $X$ defined on a prospect iff the following conditions A and B are met: It is acceptable to speak of the occurrence of $\omega$ as a realization of a random variable $X$ defined on a prospect iff the following conditions A and B are met:
A) $X$ is the measurable function A) $X$ is a measurable function
$$\begin{equation} $$\begin{equation}
X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'}) X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'})
...@@ -83,42 +84,40 @@ X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'}) ...@@ -83,42 +84,40 @@ X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'})
\end{equation}$$ \end{equation}$$
where $\Omega'$ is a set of real numbered values $X$ can take and $\mathscr{F'}$ is a set of subsets of $\Omega'$. where $\Omega'$ is a set of real numbered values $X$ can take and $\mathscr{F'}$ is a set of subsets of $\Omega'$.
I.e., $\Omega$ maps into $\Omega'$ such that correspondingly each subset $A'_i \in \mathscr{F'}$ has a pre-image I.e., $\Omega$ maps into $\Omega'$ such that correspondingly each subset $A' \in \mathscr{F'}$ has a pre-image
$$\begin{equation} $$\begin{equation}
X^{-1}A'_i \in \mathscr{F} \; , X^{-1}A' \in \mathscr{F} \; ,
\end{equation}$$ \end{equation}$$
which is the set $\{\omega_i \in \Omega: X(\omega_i) \in A'_i\}$ [@kolmogorovFoundationsTheoryProbability1950, p. 21]. which is the set $\{\omega \in \Omega: X(\omega) \in A'\}$ [@kolmogorovFoundationsTheoryProbability1950, p. 21].
B) The mapping is such that $X(\omega_i) = \omega_i$. B) The mapping is such that $X(\omega) = \omega$.
Given conditions A and B, we denote any occurrence of $\omega_i$ as a *single sample*, or realization, of a random variable defined on a prospect and any systematic approach to generate, in discrete time, a sequence of single samples that originate from multiple prospects as a *sampling strategy* [see also @hillsInformationSearchDecisions2010]. Given conditions A and B, we denote any occurrence of $\omega$ as a *single sample*, or realization, of a random variable defined on a prospect and the process of generating a sequence of single samples as *sampling*.
Note that, since single samples of a sequence originating from the same prospect are independent and identically distributed (iid), the weak law of the large number applies to the relative frequency of occurrence of an outcome $\omega$ in such a sequence [cf. @bernoulliArsConjectandiOpus1713].
Thus, large sample sequences in principle allow to obtain the same information about a prospect by sampling as by symbolic description.
## A Stochastical Sampling Model for DfE ## A Stochastical Sampling Model for DfE
Consider a choice between $1, ...,j,..., n$ prospects. Consider a choice between prospects $1, ..., k$.
To construct a rough stochastic sampling model (hereafter SSM) of the random process underlying DfE, it is assumed that agents base their decisions on the information provided by the prospects, which is in principle fully described by their probability triples. To construct a stochastic sampling model (hereafter SSM) of the random process underlying DfE, we assume that agents base their decision on the information related to these prospects and define a decision variable as a function
Thus, a decision variable
$$\begin{equation} $$\begin{equation}
D := f((\Omega, \mathscr{F}, P)_j) D:= f((\Omega, \mathscr{F}, P)_1, ..., (\Omega, \mathscr{F}, P)_k)
\;.
\end{equation}$$ \end{equation}$$
is defined. Since in DfE no symbolic descriptions of the prospects are provided, we restrict the model to the case where decisions are based on sequences of single samples originating from the respective prospects:
Since in DfE no symbolic descriptions of the triples are provided, the model is restricted to the case where decisions are based on sequences of single samples generated from the triples:
$$\begin{equation} $$\begin{equation}
D := f((X: (\Omega, \mathscr{F}) \mapsto (\Omega', \mathscr{F'}))_j) = f(X_1, ..., X_j, ..., X_n) D := f(X_{i1}, ..., X_{ik})
\; , \; ,
\end{equation}$$ \end{equation}$$
where $\Omega_j = \Omega'_j$. where $i = 1, ..., N$ denotes a sequence of length $N$ of random variables that are iid.
Note that the decision variable $D$ is defined as a function $f$ of the random variables associated with the prospects' probability spaces, where $f$ can operate on any quantitative measure, or moment, related to these random variables. Concerning the form of $f$ and the measures it utilizes, it is quite proper to say that they reflect our assumptions about the exact kind of information agents process and the way they do - not to mention whether they are capable of doing so - and that these choices should be informed by psychological or other theory [@heOntologyDecisionModels2020, for an ontology of decision models] and empirical protocols.
Since decision models differ in the form of $f$ and the measures the latter utilizes [@heOntologyDecisionModels2020, for an ontology of decision models], we take the stance that these choices should be informed by psychological or other theory and empirical protocols.
For what do these choices mean?
They reflect the assumptions about the kind of information agents process and the way they do, not to mention the question of whether they are capable of doing so.
In the following section, it is demonstrated how such assumptions about the processing strategies that agents may apply in DfE can be captured by the SSM. In the following section, it is demonstrated how such assumptions about the processing strategies that agents may apply in DfE can be captured by the SSM.
## Integrating sampling and decision strategies into the SSM ## Integrating sampling and decision strategies into the SSM
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